Unified theory of inertial granular flows and non-Brownian suspensions

Rheological properties of dense flows of hard particles are singular as one approaches the jamming threshold where flow ceases both for aerial granular flows dominated by inertia and for over-damped suspensions. Concomitantly, the length scale characterizing velocity correlations appears to diverge at jamming. Here we introduce a theoretical framework that proposes a tentative, but potentially complete, scaling description of stationary flows. Our analysis, which focuses on frictionless particles, applies both to suspensions and inertial flows of hard particles. We compare our predictions with the empirical literature, as well as with novel numerical data. Overall, we find a very good agreement between theory and observations, except for frictional inertial flows whose scaling properties clearly differ from frictionless systems. For overdamped flows, more observations are needed to decide if friction is a relevant perturbation. Our analysis makes several new predictions on microscopic dynamical quantities that should be accessible experimentally.

Figure 1

Illustration of solid destabilization: several weak contacts, indicated by red dashed lines, are opened. This induces a space of extended, disordered floppy modes, one of which is shown (arrows). Line thickness indicates force magnitude in the original, stable solid.

Figure 2

Extended vs localized contacts. When a contact is opened from an isostatic packing, the resulting deformation (arrows) can either be extended, as shown at left, or localized, as shown at right. Localized contacts are more numerous, but only extended contacts couple strongly to an imposed shear stress. Reproduced from Ref. [29] by permission of The Royal Society of Chemistry (RSC).

Figure 3

Outline of logical relationships between main macroscopic and microscopic quantities, showing the key role that $\mathcal{L}$ and $\delta z$ have in relating control parameters to the shear rate. Dashed lines indicate arguments that use an ansatz of flowing configurations being similar to destabilized isostatic ones, whereas solid lines indicate arguments independent of this assumption.

Figure 4

Typical stress vs strain curve for flow of rigid particles, showing intervals where stress relaxes smoothly, punctuated by instantaneous collisions (vertical segments).

Figure 5

Numerical verification of scaling relations in the ASM. (a) Stress anisotropy $\mu$ vs viscous number $\mathcal{J}$. Theory predicts an exponent $0.35$. (b) Volume fraction $\varphi$ vs viscous number $\mathcal{J}$. Theory predicts an exponent $0.35$. (c) Coordination deficit $\delta z$ vs viscous number $\mathcal{J}$. Theory predicts an exponent $(1+{\theta }_e)/(4+2{\theta }_e)\approx 0.30$.

Figure 6

Numerical verification of kinematic scaling relations in the ASM. (a) Lever amplitude $\mathcal{L}$ vs viscous number $\mathcal{J}$. Theory predicts an exponent $-1/2$. (b) Relaxation of lever amplitude in between collisions. Theory predicts $d\mathcal{L}/d\gamma \sim -{\mathcal{L}}^3$. [(c) and (d)] Autocorrelation of relative velocities $C\left(\gamma \right)$ vs strain $\gamma$. Panel (d) shows a scaling collapse of $C\left(\gamma \right)$, indicating the presence of a strain scale ${\epsilon }_{\eta }\sim {\mathcal{L}}^{-2}$, as predicted.

Figure 7

Divergence of suspension viscosity as measured in experiments of Ovarlez et al. [18] and Bonnoit et al. [54] (symbols), compared to our prediction $\eta \sim \delta {\varphi }^{-2.83}$ (solid). In these works the best-fit exponent was $\approx 2$ when fitted on a large range of packing fraction. However, the prediction $2.83$ appears consistent with data close enough to ${\varphi }_c$.